Application of Game Theory to Power and Rate Control for Wireless Data

Transcription

1 Application of Game Theory to Power and Rate Control for Wireless Data by Mohammad Suleiman Hayajneh B.S., Electrical Engineering, Jordan University of Science and Technology, 1995 M.S., Electrical Engineering, Jordan University of Science and Technology, 1998 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Engineering The University of New Mexico Albuquerque, New Mexico July, 2004

2 c 2004, Mohammad Suleiman Hayajneh iii

3 Dedication To my dear teacher, Professor Chaouki T. Abdallah and to my parents, sisters and brothers for their continuous inspiration iv

4 Acknowledgments I would like to thank my advisor, Professor Chaouki Abdallah, for his support, continuous guidance and inspiration and for introducing me to the field of this interesting research. I would also like to thank the dissertation committee: Professor Majid Hayat, Professor Christos Christodoulou and Professor Vladimir Koltchinskii for their insightful comments to make this research a better one. Deep thanks go to my friends: Majid Khodier, Moad Mowafi and Ousseini Lankoande for their support and encouragement. v

5 Application of Game Theory to Power and Rate Control for Wireless Data by Mohammad Suleiman Hayajneh ABSTRACT OF DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Engineering The University of New Mexico Albuquerque, New Mexico July, 2004

6 Application of Game Theory to Power and Rate Control for Wireless Data by Mohammad Suleiman Hayajneh B.S., Electrical Engineering, Jordan University of Science and Technology, 1995 M.S., Electrical Engineering, Jordan University of Science and Technology, 1998 Ph.D., Engineering, University of New Mexico, 2004 Abstract The new generations of CDMA communications systems (3G, B3G and 4G) and adhoc networks are expected to support multirate services (multimedia applications, , Internet, etc.) in addition to telephone service (fixed-rate service), which was the only service offered by 1G and 2G. Each user in these new generations of communications systems has different quality of services (QoSs) (e.g., signal-tointerference ratio (SIR), frame error rate (FER) and data rate) that he/she is willing to fulfill by accessing the common radio interface. These new services establish a strong need for new algorithms that enable the efficient spectral use of the common radio interface. Due to the strong relation between the SIR and the data rate at which a user can send information as was shown by Shannon, it was natural to propose joint power vii

7 and rate control algorithms for wireless data. Prior work that has emerged to address this problem has used a centralized algorithm. Our approach to solve the problem of jointly optimizing rate and power for wireless data in game-theoretic framework relies on two layered games: Game G1 is a non-cooperative rate control game with pricing (NRGP). It sets the rules for the users to enable them to reach a unique rate Nash equilibrium (NE) operating point that is the most socially desired operating point (Pareto efficient)in a distributed fashion. Game G2, on the other hand, is a non-cooperative power control game with pricing (NPGP). G2 admits a unique power Nash equilibrium operating point that supports the resulting rate Nash equilibrium point of game G1 with lowest possible transmit power level (Pareto efficient). In this dissertation we also propose two new distributed games: New NPGP game to optimize the transmit power for wireless data in CDMA uplink. With the rules of this game, mobile users were able to achieve higher than their minimum required SIRs (signal-to-interference ratio) with a reasonable small transmit power levels as compared to other existing NPGP games. New NPGpr game to minimize the fading induced outage probability in an interference limited wireless channels by maximizing certainty-equivalent-margin(cem) under Nakagami and Rayleigh channels. Analysis of this NPG game shows that under Rayleigh and Nakagami (with fading figure m = 2) the best policy for all users is to set their transmit power to the minimum level. Moreover, we studied the performance of NPG and NPGP games proposed by Saraydar et. al. in realistic fading wireless channels and we showed how the strategy spaces of the mobile users should be modified to guarantee the existence and uniqueness of NE operating point. viii

8 Contents List of Figures xiv List of Tables xix Glossary xx 1 Introduction What is a Game? Utility Function Non-Cooperative Game Motivation Outline Contributions Game Theoretic Power Control Algorithms in Flat-Fading Channels Utility Function and System Model ix

9 Contents 2.2 Evaluation of The Performance Rayleigh Flat-Fading Channel Rician Flat-Fading Channel Nakagami Flat-Fading Channel Non-Cooperative Power Control Game (NPG) Nash Equilibrium (NE) in NPG Non-Cooperative Power Control Game with Pricing (NPGP) Existence and Uniqueness of NE Point Simulation Results Summary Machine Learning Theory and Game Theory Utility Function and System Model NPG and NPGP Distribution-Free Learning Application to NPG and NPGP in a Slow Flat-Fading Channel Discussion of Rayleigh Slow Flat-Fading Channel and Simulation Results Summary Outage Probability In Interference Limited Wireless Fading Channels System Model x

10 Contents 4.2 Outage Probability in an Interference Limited Nakagami Flat-Fading Channel Relation Between Outage Probability and Certainty-Equivalent Margin Rayleigh Flat Fading channel Nakagami Flat Fading Channel Power Control Algorithm to Optimize The Outage Probability Simulation Results Summary New Power Control Game Theoretic Algorithms System Setup Existence of Nash Equilibrium S-Modular Games and G λ Proposed Target Function Compared to Previous Target Functions Simulation results Summary New Distributed Joint Rate and Power Control Games System Model and Our Approach Existence of Nash Equilibrium xi

11 Contents Non-Cooperative Rate Control Game with Pricing (NRGP) Non-cooperative Power Control Game with Pricing (NPGP) Simulation Results Summary Conclusions and Future work Conclusions Future Work Appendices 115 A Game Theory 116 A.1 Quasiconcavity and Quasiconvexity A.2 Standard Vector Function A.3 S-Modular Games B Channel Models 120 B.1 Rayleigh Channel B.2 Nakagami Channel B.3 Rician Channel B.4 Slow and Fast Fading Channels xii

12 Contents B.5 Frequency Selective and Frequency Nonselective Channel C Machine Learning Theory 125 C.1 Concept Learning C.2 Function Learning References 127 xiii

13 List of Figures 2.1 Equilibrium powers of NPG for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPG for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = Equilibrium powers of NPG for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPG for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = xiv

14 List of Figures 2.5 Equilibrium powers of NPGP for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPGP for Rician flat-fading channel gain (+), Rayleigh flat-fading channel gain (o), Nakagami flat-fading ( ) and deterministic channel gain (*) versus the distance of a user from the BS in meters with W/R = Equilibrium powers of NPG for Rayleigh fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPG for Rayleigh fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = Equilibrium powers of NPG for Rician fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPG for Rician fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = Equilibrium powers of NPG for Nakagami fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = xv

15 List of Figures 2.12 Equilibrium utilities of NPG for Nakagami fast flat-fading channel gain (o) and slow flat-fading channel (+) versus the distance of a user from the BS in meters with W/R = Equilibrium utilities of NPG for Rayleigh slow flat-fading channel by using (3.5.2) (o) and by simulation with samples drawn according to Rayleigh distribution (+) versus the distance of a user from the BS in meters with W/R = Equilibrium powers of NPG for Rayleigh slow flat-fading channel by using (3.5.2) (o) and by simulation with samples drawn according to Rayleigh distribution (+) versus the distance of a user from the BS in meters with W/R = Minimum equilibrium certainty-equivalent-margin in Rayleigh and Nakagami fading channels versus the threshold signal-to-interference ratio Maximum equilibrium Rayleigh fading induced outage probability ( ), the lower bound of the outage probability 1 1+CEM (solid line), the upper bound 1 e 1/CEM (dashed line) and the maximum outage probability in a Nakagami channel ( ) versus the threshold signal-tointerference ratio Equilibrium powers of the game G λ 1 ( ) and equilibrium powers of the game G λ 2 (o) versus the distance between the users and the BS with spreading gain G = 10 2 and pricing factor λ = xvi

16 List of Figures 5.2 Equilibrium SIRs of the game G λ 1 ( ) and equilibrium SIRs of the game G λ 2 (o) versus the distance between the users and the BS with spreading gain G = 10 2, pricing factor λ = 10 2 and transmission rates R i = 10 4, i N Equilibrium powers of the game G λ 1 ( ) and equilibrium powers of the game G λ 2 (o) versus the distance between the users and the BS with spreading gain G = 10 3, pricing factor λ = 10 1 and transmission rates R i = 10 4, i N Equilibrium SIRs of the game G λ 1 ( ) and equilibrium SIRs of the game G λ 2 (o) versus the distance between the users and the BS with spreading gain G = 10 3, pricing factor λ = 10 1 and transmission rates R i = 10 4, i N Equilibrium powers of the game G λ 1 ( ) and equilibrium powers of the game G λ 2 (o) versus the distance between the users and the BS with spreading gain G = 10 3, pricing factor λ = 10 1 and transmission rates R i = 10 4, i N Equilibrium SIRs of the game G λ 1 ( ), equilibrium SIRs of the game G λ 2 (o) and the minimum desired SIRs of the users ( ) versus the distance between the users and the BS with spreading gain G = 10 3, pricing factor λ = 10 1 and transmission rates R i = 10 4, i N Equilibrium powers of the game G λ 1 ( ) and equilibrium powers of the game G λ 2 (o) versus the distance between the users and the BS with spreading bandwidth W = 10 7, pricing factor λ = 10 1 and transmission rates R = [ 10 4, 10 3, 10 3, 10 2, 10 1, 10 5, 10 5, 10 5, 10 3 ] xvii

17 List of Figures 5.8 Equilibrium SIRs of the game G λ 1 ( ), equilibrium SIRs of the game G λ 2 (o) and the minimum desired SIRs of the users ( ) versus the distance between the users and the BS with spreading bandwidth W = 10 7, pricing factor λ = 10 1 and transmission rates R = [10 4, 10 3, 10 3, 10 2, 10 1, 10 5, 10 5, 10 5, 10 3 ] Normalized equilibrium rates of the game G1 ( ) and the normalized minimum required rates of the users (+) versus the user index with pricing factor λ = 10 3 and utility factors u i = Equilibrium powers of the game G2 ( ) required to support the equilibrium rates versus the user index with pricing factor λ = 10 3 and utility factors u i = B.1 Rayleigh probability density function with different fading power B.2 Nakagami probability density function with different fading figures B.3 Rician probability density function with different values of noncentrality parameter xviii

18 List of Tables 2.1 the values of parameters used in the simulations Equilibrium values of CEM i and O i for the first 10 users using in a Rayleigh flat-fading channel Perron-Frobenius theorem and the NPG game G2 introduced in this Chapter at SIR th = Equilibrium values of CEM i and O i for the first 10 users in a Rayleigh flat-fading channel using Perron-Frobenius theorem and the NPG game G2 introduced in this Chapter at SIR th = Equilibrium values of CEM i and O i for the first 10 users in a Nakagami flat-fading channel using Perron-Frobenius theorem and the NPG game G2 introduced in this Chapter at SIR th = Equilibrium values of CEM i and O i for the first 10 users in a Nakagami flat-fading channel using Perron-Frobenius theorem and the NPG game G2 introduced in this Chapter at SIR th = the values of parameters used in the simulations xix

19 Glossary p i transmit power level of user i in Watts d i distance between the BS and user i in meters h i path gain between user i and the BS P i power strategy space of the ith user p i max maximum transmit power level in P i p i min minimum transmit power level in P i p i transmit power vector of all users except for the ith user R i transmission rate (bits/s) of the ith user u i utility factor of the ith user λ; c pricing factor chosen by the BS γ i attained SIR of the ith user at the BS W spread spectrum bandwidth or the chip rate in Hz G i spreading gain of user i xx

20 Glossary N number of users currently served in the cell σ 2 variance of the background AWGN at the receiver in the BS L i target function of the ith user to be optimized NPG NPGP NRGP N Non-cooperative power control game Non-cooperative power control game with pricing Non-cooperative rate control game with pricing indexing set of the users in the cell p o i optimizing transmit power level (watts) of user i I i sum of received powers from all users except the ith user I sum of received powers from all users and AWGN power γ i minimum required SIR of user i γ vector of all minimum required SIRs of all users p ϕ i (p i, p i ) S Nash equilibrium operating point arbitrary scalar function equilibria set of an S-modular game xxi

21 Glossary p s, p l smallest and largest elements in the set S, respectively φ(p) δ arbitrary vector function scalar larger than one J i (p i, p i ) cost function in [26] p i minimizer transmit power u vector of all utility factors of users in the cell G λ 2 NPGP game with pricing factor λ α i,j fading coefficient between users j and transmitter i M L t c f c number of bits in a packet (frame) number of information bits in a frame Coherence time of the fading channel Coherence bandwidth of the fading channel T m Multipath delay spread B d Doppler spread of multipath channel TX RX BS QoS Transmitter Receiver Base station Quality of service xxii

22 Glossary CDMA DS SIR MS Code division multiple access Direct sequence Signal-to-interference ratio Mobile station xxiii

23 Chapter 1 Introduction The demand for high data rates in modern wireless CDMA communications systems, which support multirate services, increases the need to efficiently use the available radio channel bandwidth as the shared resource by the network users. The relationship between the signal-to-interference ratio (SIR) and channel capacity, as shown by Shannon [25], inspired many centralized schemes of balancing a target SIR for all users using the communication system [11]-[14]. Because of the difficulty in implementing centralized power control algorithms, and to avoid the extensive number of control signals that cause delays in the system operation, a need for distributed algorithms arose. In earlier distributed algorithms that does not use game-theoretic framework, each user is expected to allocate his own power iteratively based on local measurements to meet SIR constraints. In general, these algorithms result in large transmitter power requirements. Such algorithms may be found in [1]-[4]. To find distributed algorithms that efficiently use the transmit power, game theory was proposed. In a game-theoretic distributed power control algorithm, each user efficiently chooses his transmit power level in an attempt to optimize a target 1

24 Chapter 1. Introduction function. This target function maps the preferences and desires of the user (e.g., SIR, FER, data rate, etc.) into the real line. Such distributed algorithms are found in [5]-[10]. The mathematical theory of games was introduced by John Von Neumann and Oskar Morgenstern in 1944 [17]. In the late 1970 s game theory became an important tool in the analyst s hand whenever he or she faces a situation in which a player s decision depends on what the other players did. Game theory has been used by economists for long time to study how rational individuals interact to reach their goals. Our focus in this research will be on non-cooperative games which is a subclass of game theory. A player in a non-cooperative game, responds individually to the actions of other players by choosing a strategy from his strategy space in an attempt to optimize a target function that quantifies its QoS. The power control problem for wireless data CDMA systems was first addressed in the game theoretic framework in [5], then in a more detailed manner in [6] and [8]. The reason that game theory attracts researchers in the power control field is that it offers a good insight into the strategic interactions between rational agents (cellular users), and generates efficient outcomes according to the players preferences [19], [29]. In the remainder of this Chapter we will introduce some of the vocabulary of game theory that appear in the dissertation. 1.1 What is a Game? A game is a situation where a rational agent s decision or choice that is selected from his action profile to maximize his pay off, depends on the actions and decisions of 2

25 Chapter 1. Introduction the other rational agents. And all these rational agents have potential conflicting objectives. Therefore, the elements of a game are : A group of rational players Strategy spaces from which players choose their actions A functional description of player s preferences Strategic interdependence, that is a player s decision depends on others decisions In this research, we shall concentrate on one type of games, that is the non-cooperative game, since it represents the best game theoretic framework for the distributed power and rate control algorithms. 1.2 Utility Function A player in a game has objectives and preferences which are combined into what is called a utility function by the economists. A utility function quantifies the level of satisfaction a player obtains by adopting a strategy from his action (strategy) profile. Mathematically, a utility function maps the preferences and goals of a player into the real numbers. One should note, that the utility function is not unique, since any function that puts all the elements of the game in a desired order is a candidate function. For this reason, we may use different utility functions in the various Chapters of this dissertation. 3

26 Chapter 1. Introduction 1.3 Non-Cooperative Game A non-cooperative game is a game in which each player decides, based on local information, his strategy in response to the other players decisions. The functional description of the player s preferences (utility function) in this kind of game should be designed such that the best responses of all players converge to a fixed operating point. This operating point is known as a Nash equilibrium point as introduced by John Nash in 1951 [43]. A Nash equilibrium point is interpreted as the operating point where no player can unilaterally improve his utility by changing his strategy (on the individual s level it is the best point). Unfortunately, Nash equilibrium point may not be the most desired social point, that is the point seen by all users. The most desired social point is called a Pareto optimal point in game theory [29]. The interpretation of Pareto optimal point is that no player can improve his utility by adopting a different strategy without harming at least one other player. In other words, a Pareto optimal point maximizes the aggregate utility of all players in the game. To improve the performance of a Nash equilibrium point in non-cooperative games, a modified version of non-cooperative games was suggested by economists. In this modified version, a pricing technique on the game resources was introduced. Such modified non-cooperative game is called non-cooperative game with pricing. Any improvement in the Nash equilibrium point performance of a non-cooperative game with pricing, is called Pareto dominance with respect to the performance of Nash equilibrium point of pure non-cooperative game. 4

27 Chapter 1. Introduction 1.4 Motivation In wireless data systems that are expected to support multirate services (multimedia applications, Internet, etc.), users may desire to have a different SIRs at their receivers coupled with the lowest possible transmit powers. The importance of having a high SIR in such systems results from the need of a low error rate, a more reliable system, and high channel capacity, which allow users to transmit at higher bit rates [11],[15]. It is also important to decrease the transmit power because low-power levels help alleviate the ever present near-far problem in CDMA systems [16] and increase the lifetime of the battery in a mobile unit or in a node of adhoc networks. To simultaneously achieve these two goals many papers have emerged within the game-theoretic framework [6]-[8], or outside such framework [1]-[4]. Our focus in this research will be on power-control algorithms and joint rate and power control algorithms within a game-theoretic framework. In [8] the authors proposed asynchronous distributed algorithms for uplink CDMA wireless data in a single cell. The authors did not, however, consider the statistical variation of the power in a realistic wireless channel, as they only considered an additive-white-gaussian noise (AWGN) channel. Wireless channels are known to exhibit multipath fading, and multipath wireless channels experience two kinds of fading: Large-scale and small-scale fading. Large-scale fading results if the distance between the transmitter and the receiver is relatively large and the main contribution of the received signal comes from the reflections of the transmitted signal. The large-scale fading parameter is modeled as a log normal random variable, and it is sometimes called log-normal shadowing. Small-scale fading, on the other hand, occurs in heavily populated urban areas with a short distance between the transmitter and the receiver, and the main contribution of the received signal comes from the scattering of the transmitted signal. The small-scale fading parameter is usually 5

28 Chapter 1. Introduction modeled as Rayleigh, Rician, or Nakagami random variables. In this research we examine the algorithms of [8] in a flat-fading channel (see Appendix B for the description of flat-fading channels) and modify the algorithms and the strategy spaces to fit the flat-fading channel model. In [26], the authors proposed a cost (utility) function, which is the difference between a utility function and a pricing function. The proposed utility function is proportional to the capacity of the channel, while the pricing function is linear in the user s transmit power in market-based pricing scheme. The existence and uniqueness of a Nash equilibrium point was established in [26], and moreover, the authors offered different schemes of pricing and two methods of updating the user s transmit power. One limitation, however, of the cost function in [26] is that under market-based pricing, and if the users desire SIRs such that their utility factors are the same, the transmit power level will increase as the mobile comes closer to the base station (BS). Another limitation of the proposed algorithms in [26], is that they result in an unnecessarily high power at equilibrium. In light of the two limitations mentioned above, we propose a new target (utility or cost) function that helps the users of CDMA systems, or adhoc networks, to operate on equilibrium points that support the lowest possible power with guaranteed different QoSs for the different users. Each user in the new CDMA wireless generations and adhoc wireless networks has unique QoS requirements. Therefore, there must be realistic algorithms that take care of supporting the needs of each user in the network. To adopt more realistic (multi-objective) algorithms we study a joint power and rate control algorithm for wireless data in a game-theoretic framework. Invoking the rate in the joint optimization problem provides a fairness criterion for the power control algorithm. In other words, we need to guarantee that the obtained SIRs at equilibrium are enough for 6

29 Chapter 1. Introduction all transmitters to establish a communication link with their corresponding receivers at the required data rate. 1.5 Outline The dissertation is organized as follows: Chapter 2 is devoted to studying and modifying the algorithms in [8] in a fast/slow flat-fading wireless channels. Small-scale fading is studied using three models: Rayleigh, Rician and Nakagami channel models. In Chapter 3 we apply statistical learning theory to overcome the lack of prior knowledge of the channel model, where we learn the utility function class under a slow flat-fading channel model. As an example to validate our analysis, we show a successful application of distribution-free learning theory when the channel is modeled as a Rayleigh slow flat-fading channel. Chapter 4 presents a distributed algorithm of optimizing the outage probability for CDMA system users in an interference-limited fading wireless channel. In Chapter 5 we propose a new power control algorithm that results in a low transmit powers compared to existing algorithms. Chapter 6 presents a study of game-theoretic jointly power and rate control for wireless data. In Chapter 7, we conclude our results in this dissertation and give our thoughts of possible related future work. 1.6 Contributions In this section we list our contributions in this dissertation as follows: Non-cooperative power control games (NPG) and non-cooperative power control games with pricing (NPGP) in [8] were extended for a realistic wireless channels 7

30 Chapter 1. Introduction A distribution-free learning theory was applied to evaluate the performance of NPG and NPGP in slow flat-fading wireless channels We combined distribution-free learning theory and game theory to study the performance of NPG and NPGP algorithms for wireless data Successful NPG was proposed to minimize the outage probability in an interference limited multicell CDMA network Successful new utility function was proposed for wireless CDMA uplink which results in a very low power compared to existing ones. Introducing the first practical game-theoretic joint power and rate control for wireless data. 8

31 Chapter 2 Game Theoretic Power Control Algorithms in Flat-Fading Channels We consider in this chapter a game-theoretic power control algorithm for wireless data in a fading channel. This algorithm depends on an average utility function that assigns a numerical value to the quality of service (QoS) a user gains by accessing the channel. We also study the performance of the game-theoretic power algorithms introduced by [8] for wireless data in the realistic channels: (a1) Fast flat-fading channel and (a2) Slow flat-fading channel. The fading coefficients under both (a1) and (a2) are studied for three appropriate small-scale channel models that are used in CDMA cellular systems: A Rayleigh channel, a Rician channel and a Nakagami channel. Our results show that in a non-cooperative power control game (NPG) the best policy for all users in the cell is to target a fixed signal-to-interference and noise ratio (SIR) similar to what was shown in [8]. The difference, however, is that the target SIR in fading channels should be much higher than that in a nonfading channel. 9

32 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels The remaining of this chapter is organized as follows: In Section 2.1 we present the utility function and the system model studied in this chapter. In Section 2.2 we evaluate the performance of the system using the channel models mentioned above. Non-cooperative power control game (NPG) and Non-cooperative power control game with pricing (NPGP) are discussed in Sections 2.3 and 2.4, respectively. Then we establish the existence and uniqueness of Nash equilibrium points for NPG and NPGP under the assumed channel models in Section 2.5. Simulation results are outlined in Section 2.6. Finally, we summarize our results in Section Utility Function and System Model We use the concept of a utility function to quantify the level of satisfaction a player can get by choosing an action from its strategy profile given the other players actions, that is, a utility function maps the player s preferences into the real line. A formal definition of utility functions is available from [29]. Definition A function u that assigns a numerical value to the elements of the action set A, u : A R is a utility function if for all a, b A, action a is at least as preferred compared to b if u(a) u(b). In a cellular CDMA system there are a number of users sharing a spectrum and the air interface as a common radio resource. Henceforth, each user s transmission adds to the interference of all users at the receiver in the base station (BS). Each user desires to achieve a high quality of reception at the BS, i.e., a high SIR, by using the minimum possible amount of power to extend the battery s life. The goal of each user to have a high SIR at the BS produces conflicting objectives that make the framework of game theory suitable for studying and solving the problem. 10

33 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels In this chapter we consider a single-cell DS-CDMA (direct sequence code division multiple access) system with N users, where each user transmits frames (packets) of M bits with L information bits (M L are parity bits) [8]. The rate of transmission is R bits/sec for all users. Let P c represents the average probability of correct reception of all bits in the frame at the BS, in other words, P c refers the average frame (packet) correct reception rate, and p represents the average transmit power level. As we know, P c depends on the SIR, the channel characteristics, the modulation format, the channel coding, etc. A suitable utility function for a CDMA system is given by (see [8] and references therein): u = L R M p P c (2.1.1) where u thus represents the number of information bits received successfully at the BS per joule of expanded energy. With the assumption of no error correction, the random packet correct reception rate P c, where P c = E[ P c ], is then given as P c = M l=1 (1 P e (l)), where P e (l) is the random bit error rate (BER) of the lth bit at a given SIR γ i (c. f. (2.2.11), (2.2.27) and (2.2.45)). We are assuming that all users in a cell are using the same modulation scheme, namely non-coherent binary frequency shift Keying (BFSK), and that they are transmitting at the same rate R. 2.2 Evaluation of The Performance In this Section we find closed-form formulas of the average BERs and the average utility functions under the six assumed channel models. We then use these formulas to study the existence and uniqueness of Nash equilibrium points in Section

34 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels The SIR γ i at the receiver for the ith user is given by [15]: γ i = W R p i h i αi 2 N k i p, (2.2.1) k h k αk 2 + σ2 where α i is the path fading coefficient between ith user and the BS and it is a random variable that changes independently from bit to bit in a fast flat-fading channel (a1). On the other hand, it changes independently from packet/frame to packet/frame in a slow flat-fading channel (a2). In both cases: (a1) and (a2), the fading coefficients among the different users are assumed to be independent. The parameter W is the spread spectrum bandwidth, p k is the transmitted power of the kth user, h k is the path gain between the BS and the kth user, and σ 2 is the variance of the AWGN (additive-white-gaussian-noise) that represents the thermal noise in the receiver. For simplicity let us express the interference from all other users (i.e., all but user i) as x i, where x i = N p k h k αk 2 (2.2.2) k i therefore (2.2.1) may be written as: γ i = W R := γ iα 2 i p i h i x i + σ 2 α2 i (2.2.3) For a given α i and x i, the conditioned BER, P (e γ i ) = P (e γ i, x i ) (the dependence on x i comes through γ i ), of the ith user using non coherent BFSK is given by [15]: P (e γ i, x i ) = 1 2 e γ i 2 (2.2.4) The average BER and average utility functions for this modulation scheme are evaluated in this chapter for the following channel models: Rayleigh fast/slow flat-fading channel, Rician fast/slow flat-fading channel and Nakagami fast/slow flat-fading channel. 12

35 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Rayleigh Flat-Fading Channel In this case α i is modeled as a Rayleigh random variable with a probability density function (PDF) given by (see Appendix B for the description of channel models): f α i (ω) = ω σ 2 r e (1/2σ2 r)ω 2, i = 1, 2,, N (2.2.5) In all following calculations, and as a consequence of the multiplicative channel model of small and large scale fading, it is assumed that σ 2 r = 1/2. Using (2.2.3) and (2.2.5) the PDF of γ i for a given x i is defined as: f γ i x i (ω) = 1 γ i e ω/γ i (2.2.6) Rayleigh Fast Flat-Fading Channel For the lth bit in the frame, we can rewrite the SIR (2.2.3) and the interference (2.2.2) for the ith user as follows: γ i (l) = W R p i h i α 2 i (l) x i (l) + σ 2 (2.2.7) x i (l) = N p k h k αk(l) 2 (2.2.8) k i Assuming that both {α i (l)} M l=1 and {x i(l)} M l=1 are iid (identically independent distributed) random variables, and of course α i (l) and x i (l) are independent random variables. Henceforth, the averaged correct reception of all frame (packet) bits at the BS P c is given as (1 P e ) M, where P e is averaged BER for each bit in the frame, that is P e = E[ P e ]. We will calculate the averaged P e next. We can find the conditional error probability P (e x i ) by taking the average of 13

36 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels (2.2.4) with respect to f γ i x i (ω): [ ] P (e x i ) = E P (e γi, x i ) = 0 = 1 2γ i 1 = 2 + γ i P (e ω, x i ) f γ i x i (ω)dω 0 2+γ ( i 2γ )ω i dω e (2.2.9) Notice that we dropped the bit index l because the average BER does not depend on l. For large SIR, (2.2.9) behaves as: P (e x i ) 1 γ i = x i + σ 2 W R p i h i (2.2.10) Now, we can find the averaged BER P e by taking the expectation of (2.2.10): [ ] P e = E P (e x i ) = E[x i] + σ 2 W p R i h i = 1 γ i (2.2.11) where γ i is the ratio of the mean of the received power from user i to the mean of the interference at the receiver as given by: γ i = W R p i h i N k i p k h k + σ 2 (2.2.12) Therefore, the average utility function of the ith user is given by: u i = L R M p i (1 1 γ i ) M (2.2.13) Rayleigh Slow Flat-Fading Channel In a slow flat-fading channel model, α i of the ith user is assumed to change independently for each packet/frame, that is α i (1) = α i (2),, α i (M). Also, it is assumed 14

37 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels that α i and α k are independent for all i k. The averaged frame correct reception P c is therefore given as the expectation of (1 P (e γ i, x i )) M with respect to the random variables γ i and x i. This suggests rewriting (2.1.1) as: u i (p γ i, x i ) = L R M p i (1 e γ i/2 ) M (2.2.14) Where p = (p 1, p 2,, p N ) is the vector of transmit powers of all users. Note that P e was replaced by 2 P e to give the utility function u i (p γ i, x i ) this property: u i (p γ i, x i ) 0 as p i 0 and u i (p γ i, x i ) 0 as p i [8]. One can evaluate u i (p x i ) as follows: u i (p x i ) = = = 0 0 L R u i (p ω, x i ) f γ i x i (ω) dω L R (1 e ω/2 ) M M p i M M p i γ i M = L R M p i k=0 k=0 ( 1) k ( M k ( M k ) 2 ( 1) k 1 e ω/γ γ i dω i ) 0 e ( k γ ) ω i dω k γ i + 2 (2.2.15) For γ i 1, (2.2.15) can be approximated by: ( u(p x i ) L R M p i γ i M k=1 ( M k ) ) 2 ( 1) k k (2.2.16) 15

38 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Averaging (2.2.16) with respect to x i we obtain the average utility function for high SIR: u i = E [u i (p x i )] ( L R 1 + E[x i] + σ 2 M ( ) ) M 2 ( 1) k W M p i p R i h i k k k=1 ( = L R M ( ) ) M 2 ( 1) k M p i γ i k k k=1 u i L R ) (1 βγi M p i (2.2.17) where β = M ( M ) 2 ( 1) k k=1 > 0. k k Rician Flat-Fading Channel In this case, α i is modeled as a Rician random variable with PDF given by (see Appendix B): f α i (ω) = ω σ 2 r e ( ω 2 +s 2 2σr 2 ) I0 ( ω s ) (2.2.18) σr 2 Similarly to the Rayleigh case, we need to find the PDF of γ i (see (2.2.3)) for fixed x i (see (2.2.2)): f γ i x i (ω) = e s2 γ i e ω/γ i ω I 0 (2s ) (2.2.19) γ i where we assumed that σ 2 r = 1/2 as we mentioned earlier. 16

39 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Rician Fast Flat-Fading Channel Similar to the Rayleigh fast flat-fading case, the averaged frame correct reception is given by P c = (1 P e ) M, where P e can be found as follows: P (e x i ) = 0 = e s2 2γ i P (e ω, x i )f γ i x i (ω)dω ( ) e ω( γ ) ω i I0 2s dω (2.2.20) 0 γ i using the fact that I 0 (ζ) can be written as: I 0 (ζ) = n=0 ( ζ 2 )2n (n!) 2 (2.2.21) and substituting (2.2.21) in (2.2.20), and after few mathematical manipulations we obtain: P (e x i ) = γ i e s2 ( γ ) i (2.2.22) At high SIR (γ i 1), P (e x i ) may be approximated as: P (e x i ) 1 γ i e s2 = x i + σ 2 W R p i h i e s2 (2.2.23) In order to find the final average error rate P e, we need to find µ x i the mean of x i. [ N ] µ x i = E[x i ] = E αkp 2 k h k k i N = p k h k E[αk] 2 = (1 + s 2 ) k i k i N p k h k (2.2.24) where we used the fact that [15] E[αk] n = (2σr) 2 n/2 e ( s 2 2σr 2 ) Γ((2 + n)/2) Γ(n/2) 1F 1[(2 + n)/2, n/2; s 2 /2σ 2 r] (2.2.25) 17

40 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels where Γ(.) is the Gamma function, and 1F 1[a, b; y] is the confluent hypergeometric function [20]. By substituting for σr 2 = 1/2 and n = 2 in (2.2.25) we obtain the result in (2.2.24). Note that we have used the following special case of the confluent hypergeometric function 1F 1[a, b; y] in calculating (2.2.24): 1F 1[2, 1; s 2 ] = (1 + s 2 ) e s2 (2.2.26) Finally, to obtain P e we replace x i in (2.2.23) by µ x i, that is P e e s2 (µ x i + σ 2 ) W R h i p i = 1 γ s i (2.2.27) where γ s i = e s2 W h R i p i (1 + s 2 ) N k i h (2.2.28) k p k + σ 2 Then, the utility function of the ith user is given by u i = L R (1 1 ) M (2.2.29) M p i γi s Rician Slow Flat-Fading Channel Following the same argument of the Rayleigh slow flat-fading case, we find the average P c or equivalently the average utility function of the ith user as follows. u i (p x i ) = = 0 0 u i (p ω, x i )f γ i x i (ω)dω L R M p i (1 e ω/2 ) M e s2 γ i e ω/γ i ω I 0 (2s ) dω (2.2.30) γ i By substituting (2.2.21), factorizing (1 e γ i/2 ) M, and after few mathematical manipulations we obtain: u i (p x i ) = L R M p i M ( ) M ( 1) k k k=0 2 es2( 1+ 2 k γ ) k γ i + 2 i +2 (2.2.31) 18

41 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels For large SIR, γ i 1 the above equation can be approximated by: [ u i (p x i ) L R 1 + e s2 M p i γ i M ( M ( 1) k k k=1 ) ] 2 k (2.2.32) Averaging (2.2.32) with respect to x i we obtain the final approximate averaged utility function of the ith user in the following form: [ u i L R M p i γi s u i L R M p i M ( M ( 1) k k k=1 [ 1 β γ s i ] ) ] 2 k (2.2.33) with γ s i given by (2.2.28) Nakagami Flat-Fading Channel In this case α i is modeled as a Nakagami random variable with PDF given by (see Appendix B) [15]: f α i (ω) = 2m m Γ(m)Ω m ω2m 1 e ( m Ω )ω2, (2.2.34) Note that by setting m = 1 the Nakagami PDF reduces to the Rayleigh PDF. In the following calculations it is assumed that Ω = 1. Then the PDF of γ i for fixed x i is given by: f γ i x i (ω) = 1 Γ(m) ( ) m m ω m 1 e ( m γ i γ ) ω i (2.2.35) 19

42 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Nakagami Fast Flat-Fading Channel Here, we find the conditioned error probability P (e x i ) by taking the average of (2.2.4) with respect to f γ i x i (ω): P (e x i ) = = 0 P (e ω, x i ) f γ i x i (ω)dω ( ) m 1 m ω m 1 e ( γ i +2m 2γ ) ω 2Γ(m) γ i dω i 0 ( ) m 2m (2.2.36) = 1 2 2m + γ i For fixed m and γ i 1, (2.2.36) can be rewritten as: ( ) m 2m (2.2.37) P (e x i ) 1 2 γ i To find the average P e, we need to find the mean of (x i + σ 2 ) m. Here, x i is the sum of independent random variables, each distributed according to a Gamma density function. This makes the evaluation of (x i + σ 2 ) m a tedious mathematical job. In such a case it is easier to find an approximate density function of x i. To do this, we recall Esseen s inequality which estimates the deviation of the exact PDF of a sum of independent variables from the normal PDF. Theorem [21] let Y 1,, Y N be independent random variables with E[Y j ] = 0, E[ Y j 3 ] < (j = 1,, N). Let σ 2 j := E[Y 2 j ], B N := N j=1 σ 2 j, L N := B 3/2 N N E[ Y j 3 ] Let ψ K (z) be the c.f. (cumulative distribution ) of the random variable B 1/2 N N j=1 Y j. Then j=1 ψ N (z) e z2 /2 16 L N z 3 e z2 /3 (2.2.38) 20

43 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Proof : See [21]. Let us define Ỹk := p k h k α 2 k and Y k := Ỹk p k h k. By simple calculations we find that Ỹk, (k = 1,, N) are Gamma distributed random variables, such that f Ỹk (ω) = (m/p kh k ) m Γ(m) ω m 1 e (m/p kh k )ω and E[Ỹk] = p k h k, which means that Y k, (k = 1,, N) are zero mean random variables. The values of σ 2 k = E[Y 2 k ] are (p kh k ) 2 /m, k = 1,, N, and therefore, B N = 1 m N k=1 (p kh k ) 2. It is fairly simple to find out that the third moment E[ Y k 3 ] = E[Y 3 k ] = 2(p kh k ) 3 m 2 (Y k 0). Then, L N = 2 N k=1 (p kh k ) 3 m ( N k=1 (p kh k ) 2 ) 3/2. For large N, L N has a very small value, i.e., L N << 1. By examining (2.2.38) for small values of z, L N takes care of righthand side of the inequality, making it very small. For large values of z, on the other hand, the exponential term e z2 /3 will decrease the bound and make it approach zero. In conclusion, we can approximate x i as a Gaussian random variable with mean ζ x i and variance σ 2 x i as given by: [ N ] ζ x i = E[x i ] = E αkp 2 k h k = = k i N p k h k E[αk] 2 k i N p k h k (2.2.39) k i 21

44 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels and σ 2 x i = E[x 2 i] ζx 2 [ i N ] N = E p l h l p k h k αl 2 αk 2 ζx 2 i = 1 m l i k i N (p k h k ) 2 (2.2.40) k i where (2.2.40) was obtained using the fact that α k and α l are statistically independent for all k l. Thus, we can write π(x i ), the PDF of x i, as follows: π(x i ) = 1 e (x i ζx i ) 2 2σ 2 x i (2.2.41) 2πσx i where x i 0. Averaging (2.2.37) over π(x i ) we obtain the average error probability P e for high SIR as follows: P e 1 2 = ( ( ( 2m W p R i h i 2m W p R i h i 2m W R p i h i ) m 0 ) m ) m ( x i + σ 2) m 1 2πσx i y m 1 σ 2 2πσx i 0 y m 1 2πσx i e (y (ζx i +σ 2 )) 2 2σ 2 x i dy e (x i ζx i ) 2 2σ 2 x i dx i e (y (ζx i +σ 2 )) 2 2σ 2 x i dy (2.2.42) where we used the change of variable y = x i + σ 2 and the last approximation in (2.2.42) was based on the fact that σ 2 1. By examining (2.2.42) one can see that it represents the mth moment of a random variable normally distributed with mean ζ y = ζ x i + σ 2 and variance σ 2 y = σ 2 x i. 22

45 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Therefore, the average P e is given by: ( ) P e = 1 m 2m W 2 p E [y m ] R i h i ( ) = 1 m 2m W 2 p E [((y ζ y ) + ζ y ) m ] R i h i ( ) = 1 m 2m m ( ) m W 2 p ζy m k C k R i h i k k=0 ( ) = 1 m 2 m ζ y m ( ) m Ck W 2 p R i h i k µ k k=0 y ( ) m m m ( ) m = 2 m 1 Ck γ i k k=0 ζ k y (2.2.43) where γ i is given in (2.2.12), and C k is the kth central moment given by [15]: 1.3 (k 1) σ k x C k = i k even 0 k odd By splitting up the summation in (2.2.43), we obtain: m l=0 ( ) m Cl l ζy l ( ) m σ 2 x = 1 + i 2 (σ 2 + N k i p k h k ) + 2 ( ) m 1.3 (m 1)σ m 1 x + i m (σ 2 + N k i p (2.2.44) k h k ) m where m = m if m is even and m = m 1 if m is odd. Since σ 2 x (see (2.2.40)) is very small compared to ζ x i (see (2.2.39)), we can approximate the summation by its leading term, namely 1. Therefore the average P e at high SIR behaves as: P e 2 m 1 ( m γ i ) m. (2.2.45) Therefore the utility function of the ith user is given as: u i = L R M p i ( 1 2 m 1 ( m γ i ) m ) M. (2.2.46) 23

46 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels Notice that if we set m = 1, we obtain the same performance as in the Rayleigh slow flat-fading case. Nakagami Slow Flat-Fading Channel As described earlier, u i (p x i ) can be determined as follows: u i (p x i ) = = 0 0 u i (p ω, x i )f γ i x i (ω)dω L R (1 e ω/2 ) M 1 M p i Γ(m) ( ) m m ω m 1 e ( m γ ) ω i dω(2.2.47) γ i By factoring (1 e γi/2 ) M and using the identity y n e a y dy = Γ(n+1) 0 a n+1 u i (p x i ) = L R M p i we obtain: M ( ) ( ) m M ( 1) k 2 m k k γ i + 2 m (2.2.48) k=0 For fixed m and high SIR, γ i 1, (2.2.48) may be approximated as: [ u i (p x i ) L R 1 + ( 1 M p i γ i M ( ) ( ) ] m M 2 m ) m ( 1) k. (2.2.49) k k k=1 Averaging (2.2.49) with respect to the distribution of x i and using the same argument as in (2.2.42), (2.2.43) and (2.2.44), we end up with the final approximate averaged utility function given by: [ u i L R 1 + ( 1 M ( ) ( ) ] m M 2 m ) m ( 1) k M p i γ i k k k=1 u i L R ] [1 ξ ( 1γi ) m M p i (2.2.50) where ξ = M k=1 ( 1)k ( M k ) ( 2 m k ) m > 0. 24

47 Chapter 2. Game Theoretic Power Control Algorithms in Flat-Fading Channels 2.3 Non-Cooperative Power Control Game (NPG) Let N = {1, 2,, N} represent the index set of the users currently served in the cell and {P j } j N represents the set of strategy spaces of all users in the cell. Let G = [N, {P j }, {u j (.)}] denote a noncooperative game, where each user, basing on local information, chooses its power level from a convex set P j = [p j min, p j max ] and where p j min and p j max are respectively the minimum and the maximum power levels in the jth user strategy space. With the assumption that the power vector p = [p 1, p 2,, p N ] is the result of NPG, the utility of user j is given as [8]: u j (p) = u j (p j, p j ) (2.3.1) where p j is the power transmitted by user j, and p j is the vector of powers transmitted by all other users. The right side of (2.3.1) emphasizes the fact that user j can just control his own power. We can rewrite (2.1.1) for user j as: u j (p j, p j ) = L R M p j P c (γ j ) (2.3.2) The formal expression for the NPG is given in [8] as: G : max u j (p j, p j ), for all j N (2.3.3) p j P j Where u j (p j, p j ) is a continuous function. This game will produce a sequence of power vectors that converges to a point where all users are satisfied with the utility level they obtained. This operating point is called a Nash equilibrium operating point of NPG. In the next subsection, we define the Nash equilibrium point and describe its physical interpretation Nash Equilibrium (NE) in NPG The resulting power vector of NPG is called a Nash equilibrium power vector. 25